A Question about Topological Connectivity

I am not a Mathematician, and I've been pondering this idea for years. I will try to describe it intelligibly.

Imagine a Ring. It has three "Inputs" and three "Outputs".

Any of the three "Outputs" takes you to a different Ring with three Entrances and three Exits.

You cannot return to the original Ring in less than three moves. Nor do any second or third level Rings connect directly to any common Rings, nor does any Ring have both a Direct Entrance and Exit to the same Ring.....

What is the minimum number of interconnecting Rings I will need, to make the system "Circular" and "Homogenous" in the sense that it always leads inevitably back to the starting point, if one avoids entering any other Ring more than once?

Is there a Maxim number of Rings that can be connected in this way?

I assume that the structure would be Symmetric, in that any Ring that I choose arbitrarily to start from, would inevitably take me Back to the starting Ring in the same number of non-retracing steps.

This thing grows to the Point of being a Wooly-Bear Worm to Visualize. Are there are Equations to "Flatten it out"--that is, give results without needing to clearly visualize it?

Believe it or not, I'm trying to create an an Abstract Three Value Symbolic Logic System, using the Interlocking Circles in Place of a "T" "F" Truth Table.

For example, you never explained what the "second and third level rings" were. Pretty much everything is unclear. Are there rules for moving through these rings that you have to follow, other than exiting from an exit and entering from an entrance?

In any case, it sounds like the right framework for it would be graph theory, where people have thought about all these types of questions for various graphs.